Lecture 3 Gauss

1
Lecture 3 Gausss Law Chp. 24

• Cartoon - Electric field is analogous to
  gravitational field
• Opening Demo -
• Warm-up problem
• Physlet /webphysics.davidson.edu/physletprob
• Topics
• Flux
• Electric Flux and Example
• Gauss Law
• Coulombs Law from Gauss Law
• Isolated conductor and Electric field outside
  conductor
• Application of Gauss Law
• Charged wire or rod
• Plane of charge
• Conducting Plates
• Spherical shell of charge
• List of Demos
• Faraday Ice pail metal cup, charge ball,
• teflon rod, silk,electroscope

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Electric Flux
Flux is a measure of the number of field lines
passing through an area. Electric flux is the
number of Electric field lines penetrating a
surface or an area.
In general,
a
b
3
Gausss Law

• Gausss law makes it possible to find the
  electric field easily in highly symmetric
  situations.
• Drawing electric field lines around charges leads
  us to Gauss Law
• The idea is to draw a closed surface like a
  balloon around any charge distribution, then some
  field line will exit through the surface and some
  will enter or renter. If we count those that
  leave as positive and those that enter as
  negative, then the net number leaving will give a
  measure of the net positive charge inside.

4
Electric lines of flux and Gausss Law

• The flux ? through a plane surface of area A
  due to a uniform field E
• is a simple product
• where E is normal to the area A .

• because the normal component of E is 0

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Approximate Flux
Exact Flux
Circle means you integrate over a closed surface.
6
Find the electric flux through a cylindrical
surface in a uniform electric field E
a.
What would be the flux if the cylinder were
vertical ?
b.
Suppose it were any shape?
c.
Flux from a. b. c. 0
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Electric lines of flux and Derivation of Gauss
Law using Coulombs law

• Consider a sphere drawn around a positive point
  charge. Evaluate the net flux through the closed
  surface.

Net Flux
For a Point charge
dA
Gauss Law
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Gauss Law

• This result can be extended to any shape surface
• with any number of point charges inside and
• outside the surface as long as we evaluate the
• net flux through it.

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Applications of Gausss Law

• Find electric filed of an infinite long uniformly
  charged wire of negligible radius.
• Find electric field of a large thin flat plane or
  sheet of charge.
• Find electric field around two parallel flat
  planes.
• Find E inside and outside of a long solid
  cylinder of charge density ? and radius r.
• Find E for a thin cylindrical shell of surface
  charge density ?.
• Find E inside and outside a solid charged sphere
  of charge density ?.

10
Electric field in and around conductors

• Inside a conductor in electrostatic
• equilibrium the electric field is zero
• ( averaged over many atomic volumes).
• The electrons in a conductor move
• around so that they cancel out any
• electric field inside the conductor
• resulting from free charges
• anywhere including outside the
• conductor. This results in a net force of
• 0 inside the conductor.

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Electric field in and around conductors

• Any net electric charge resides
• on the surface of the conductor
• within a few angstroms (10-10 m).
• Draw a Gaussian surface just
• inside
• the conductor. We know
• everywhere on this surface.
• Hence, the net flux is zero. Hence,
• the net charge inside is zero.
• Show Faraday ice pail demo.

12
Electric field in and around conductors

• The electric field just outside a conductor has
  magnitude and is directed perpendicular to
  the surface.
• Draw a small pill box that extends
• into the conductor. Since there is
• no field inside, all the flux comes
• out through the top.

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Two Conducting Plates
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Negative charge in a neutral conducting metal
shell
15
Find the electric field for an infinite long wire
Charge per unit length
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Application of Gausss Law

• Electric field inside and outside a solid
  uniformly charged sphere

• Often used as a model of the nucleus.
• Electron scattering experiments have shown that
  the charge density is constant for some radius
  and then suddenly drops off at about

For the nucleus,
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Electric Field inside and outside a uniformly
charged sphere
Inside the sphere To find the charge at a
distance rltR Draw a gaussian surface of radius
r By symmetry E is radial and parallel to normal
at the surface. By Gausss Law
Outside the sphere
Same as a point charge q
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Electric field vs. radius for a conducting sphere
(similar to gravity)